In FIG. REF , the dependence of \(s\) , \(c_s^2\) , \(C_v\)  and \(F\)  on \(T\)  for \(V_{1{\rm st}}\)  are shown. From these diagrams one can see that in the range of temperature \(T_m< T <1.272T_m\)  there are three branches of solutions where two of them correspond to the stable black hole solution (solid blue curves) and one of them corresponds to the unstable black hole solution (dashed cyan curve). There are no unstable solutions at \(T< T_m\)  and \(1.272T_m< T\) . The dashed curves show that the system indicates the Gregory-Laflamme instability [1]}, [2]} which can be seen from the behavior of \(C_v\) . In [3]}, [4]}, [5]} it is shown that in the absence of conserved charges related to gauge symmetries, the Gregory-Laflamme instability is equivalent to the negative values of \(C_v\)  which is depicted clearly in panel REF . From the panels REF  and REF  we observe that the negative values of \(C_v\)  corresponds to the negative \(c_s^2\) . \(F(T)\)  shows that the phase transition occurs at critical temperature \(T_c\simeq 1.05T_m\)  in which the two stable branches of solutions cross each other at this point. Although \(F\)  is continuous across the phase transition, \(\frac{dF}{dT}=s\)  is not continuous which means the system possesses a first order phase transition at \(T_c\) . Since \(s/T^3\)  counts the effective number of DOF, this number increases suddenly when we approach to \(T_c\)  from the left [6]}.
